Measure solution for the growth-fragmentation equation in a critical case: existence, uniqueness and convergence toward an oscillatory measure.
For decades now, structured population models have been widely studied. More recently, measure solutions to these equations drew attention of the community, for at least two reasons: first, measure solutions account for concentration phenomena, and second, allowing a delta function as an initial condition is the right way to model the initiation of a bacteria culture with a single individual.
In the first part of the talk, I will recall the two most famous models in the field of structure population models, namely the renewal equation and the growth-fragmentation equation. Then, I will define what a measure solution is, and link this definition to some recent results, both for measure and L^1 solutions, in order to motivate the study of the problem we are interested in: to prove existence, uniqueness and long time asymptotic of a measure solution to the size equation in the special case of linear growth rate and equal mitosis.
In the second part, I will present the strategy we chose to adopted to tackle this problem: by duality, we first obtain informations on the (continuous) solution to the adjoint problem through Generalized Relative Entropy technique, and then back the the primal problem, we deduce the asymptotic behavior for the measure solution.
Finally, if time allows it, I'll show a numerical scheme (the Escalator Boxcar Train) that allows us to approximate measure solutions.